This drastic change in appearance for Klingons was a bit of a running joke among fans during later series of Star Trek. The writers even acknowledged it during an episode of Deep Space Nine which sees Worf, among others, time-travel back into the original series episode The Trouble with Tribbles. When asked why those Klingons looked funny he curtly replies
We do not discuss it with outsiders.
Many fan theories popped up over the years to explain away the problem. Finally, though, we got a canon answer in Enterprise, specifically the episodes "Affliction" and "Divergence". If you want the whole story in all it's detail, I'd highly recommend watching the entire story arc (it also ties in with The Wrath of Khan among other things.)
The basic idea, though, is that the Klingons got ahold of the genetically engineered humans from the Eugenics Wars and tried to use them to create genetically engineered Klingons. Something went wrong, and a odd strain of the flu virus infected many of the subjects, changing their appearance. This mutated look remained in the Klingon gene pool for a while afterward. Eventually, a cure for the flu was found, but it was a while longer before they were able to reverse the effects.